PageRank, Connecting a Line of Nodes with a Complete Graph
The focus of this article is the PageRank algorithm originally defined by S. Brin and L. Page as the stationary distribution of a certain random walk on a graph used to rank homepages on the Internet. We will attempt to get a better understanding of how PageRank changes after you make some changes to the graph such as adding or removing edge between otherwise disjoint subgraphs. In particular we will take a look at link structures consisting of a line of nodes or a complete graph where every node links to all others and different ways to combine the two. Both the ordinary normalized version of PageRank as well as a non-normalized version of PageRank found by solving corresponding linear system will be considered. We will see that it is possible to find explicit formulas for the PageRank in some simple link structures and using these formulas take a more in-depth look at the behavior of the ranking as the system changes.
KeywordsPageRank Graph Random walk Block matrix
This research was supported in part by the Swedish Research Council (621- 2007-6338), Swedish Foundation for International Cooperation in Research and Higher Education (STINT), Royal Swedish Academy of Sciences, Royal Physiographic Society in Lund and Crafoord Foundation.
- 1.Battiston, S., Puliga, M., Kaushik, R., Tasca, P., Caldarelli, G.: DebtRank: Too central to fail? Financial networks, the FED and systemic risk. Sci. Rep. 2, 541 (2012)Google Scholar
- 2.Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. No. del 11 in Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (1994)Google Scholar
- 3.Bernstein, D.: Matrix Mathematics. Princeton University Press, Princeton (2005)Google Scholar
- 4.Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Comput. Network. ISDN Syst. 30(1–7), 107–117 (1998). Proceedings of the Seventh International World Wide Web ConferenceGoogle Scholar
- 6.Engström, C., Silvestrov, S.: A componentwise pagerank algorithm. In: Applied Stochastic Models and Data Analysis (ASMDA 2015). The 16th Conference of the ASMDA International Society (2015, in press)Google Scholar
- 7.Engström, C., Silvestrov, S.: Pagerank, a look at small changes in a line of nodes and the complete graph. Engineering Mathematics II, Springer Proceedings in Mathematics & Statistics 179 (2016)Google Scholar
- 9.Haveliwala, T., Kamvar, S.: The second eigenvalue of the google matrix. Technical report 2003-20, Stanford InfoLab (2003)Google Scholar
- 10.Ishii, H., Tempo, R., Bai, E.W., Dabbene, F.: Distributed randomized pagerank computation based on web aggregation. In: Proceedings of the 48th IEEE Conference on Decision and Control, 2009 Held Jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009, pp. 3026–3031 (2009)Google Scholar
- 11.Kamvar, S., Haveliwala, T.: The condition number of the pagerank problem. Technical report 2003-36, Stanford InfoLab (2003)Google Scholar
- 12.Kamvar, S.D., Schlosser, M.T., Garcia-Molina, H.: The eigentrust algorithm for reputation management in p2p networks. In: Proceedings of the 12th International Conference on World Wide Web. WWW 2003, pp. 640–651. ACM, New York (2003)Google Scholar
- 15.Tobias, R., Georg, L.: Markovprocesser. Univ., Lund (2000)Google Scholar